Heisenberg Uncertainty Principle

Quantum Mechanics A

Schrödinger Equation The most fundamental equation of quantum mechanics; given a Hamiltonian ${\mathcal {H}}$, it describes how a state Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\Psi\rangle} evolves in time.

Physical Basis of Quantum Mechanics

Basic Concepts and Theory of Motion UV Catastrophe (Black-Body Radiation) Photoelectric Effect Stability of Matter Double Slit Experiment Stern-Gerlach Experiment The Principle of Complementarity The Correspondence Principle The Philosophy of Quantum Theory

Schrödinger Equation

Brief Derivation of Schrödinger Equation Relation Between the Wave Function and Probability Density Stationary States Heisenberg Uncertainty Principle Some Consequences of the Uncertainty Principle

Operators, Eigenfunctions, and Symmetry

Linear Vector Spaces and Operators Commutation Relations and Simultaneous Eigenvalues The Schrödinger Equation in Dirac Notation Transformations of Operators and Symmetry Time Evolution of Expectation Values and Ehrenfest's Theorem

Motion in One Dimension

One-Dimensional Bound States Oscillation Theorem The Dirac Delta Function Potential Scattering States, Transmission and Reflection Motion in a Periodic Potential Summary of One-Dimensional Systems

Discrete Eigenvalues and Bound States

Harmonic Oscillator Spectrum and Eigenstates Analytical Method for Solving the Simple Harmonic Oscillator Coherent States Charged Particles in an Electromagnetic Field WKB Approximation

Time Evolution and the Pictures of Quantum Mechanics

The Heisenberg Picture: Equations of Motion for Operators The Interaction Picture The Virial Theorem

Angular Momentum

Commutation Relations Angular Momentum as a Generator of Rotations in 3D Spherical Coordinates Eigenvalue Quantization Orbital Angular Momentum Eigenfunctions

Central Forces

General Formalism Free Particle in Spherical Coordinates Spherical Well Isotropic Harmonic Oscillator Hydrogen Atom WKB in Spherical Coordinates

The Path Integral Formulation of Quantum Mechanics

Feynman Path Integrals The Free-Particle Propagator Propagator for the Harmonic Oscillator

Continuous Eigenvalues and Collision Theory

Differential Cross Section and the Green's Function Formulation of Scattering Central Potential Scattering and Phase Shifts Coulomb Potential Scattering

Consider a long string which contains a wave that moves with a fairly well-defined wavelength across the whole length of the string. The question, "where is the wave" does not seem to make much sense, since it is spread throughout the length of string. A quick snap of the wrist and the string produces a small bump-like wave which has a well defined position. Now the question, "what is the wavelength" does not make sense, since there is no well defined period. This example illisturates the limitation on measuring the wavelength and the position simultaneously. Relating the wavelength to momentum yields the de Broglie equation, which is applicable to any wave phenomenon, including the wave equation:

$p={\frac {h}{\lambda }}={\frac {2\pi \hbar }{\lambda }}$

Now that there is a relation between momentum and position, the uncertainty of the measurement of either momentum or position takes mathematical form in the Heisenberg Uncertainty relation:

$\Delta x\,\Delta p\geq {\frac {\hbar }{2}}$

where the $\Delta A$ of each operator represents the positive square root of the variance, given generally by:

$(\Delta A)^{2}=\langle (A-\langle A\rangle )^{2}\rangle =\langle A^{2}\rangle -\langle A\rangle ^{2}.$

Although both momentum and position are measurable quantities that will yield precise values when measured, the uncertainty principle states that the deviation in one quantity is directly related to the other quantity. This deviation in the uncertainty principle is the result of identically prepared systems not yielding identical results.

A generalized expression for the uncertainty of any two operators A and B is:

$\Delta A\,\Delta B={\frac {1}{2i}}\langle [A,B]\rangle .$

And thus, there exists an uncertainty relation between any two observables which do not commute.

More generally the uncertainty principle states that two canonically conjugated variables cannot be determined simultaneously with a precision exceeding the relation:

$\Delta A\,\Delta B=\hbar$

Canonically conjugated variables are those which are related by the Fourier Transform. More specifically, they are variables that when you take the Fourier Transform of a function that is dependent on one, you get a function that depends on the other. For example, position and momentum are canonically conjugated variables:

$\Phi (p,t)={\frac {1}{2\pi \hbar }}\int _{-\infty }^{+\infty }e^{\frac {-ipx}{\hbar }}\Psi (x,t)dx$; $\Psi (x,t)={\frac {1}{2\pi \hbar }}\int _{-\infty }^{+\infty }e^{\frac {ipx}{\hbar }}\Phi (p,t)dp$.

Another example of canonically conjugated variables are energy and time. It is precisely this relationship that leads to the uncertainty principle. The reader has probably noticed that the relation above – i.e. $\Delta A\,\Delta B=\hbar$ – is not the familiar uncertainty principle we all know, the one where $\hbar$ is divided by two. It turns out that the above relation is more general; we only get the more familiar version when the wave-packet is Gaussian.

A generalized proof of the Uncertainty Principle is as follows.

For any observable $A$, we have

$(\Delta A)^{2}=\langle ({\hat {A}}-\langle A\rangle )\Psi |({\hat {A}}-\langle A\rangle )\Psi \rangle =\langle f|f\rangle$

where $f\equiv ({\hat {A}}-\langle A\rangle )\Psi$. Likewise, for any other observable $B$,

$(\Delta B)^{2}=\langle g|g\rangle$ where $g\equiv ({\hat {B}}-\langle B\rangle )\Psi$.

Now we invoke the Schwartz inequality. Recall that this is just

$\left|\int _{a}^{b}[f(x)]^{*}g(x)dx\right|\leq {\sqrt {\int _{a}^{b}|f(x)|^{2}\,dx\int _{a}^{b}|g(x)|^{2}\,dx}}$

Invoking this expression,

$(\Delta A)^{2}(\Delta B)^{2}=\langle f|f\rangle \langle g|g\rangle \geq |\langle f|g\rangle |^{2}$.

Now, for any complex number $z$,

$|z|^{2}=[{\text{Re}}(z)]^{2}+[{\text{Im}}(z)]^{2}\geq [{\text{Im}}(z)]^{2}=\left({\frac {z-z^{*}}{2i}}\right)^{2}$.

Letting $z=\langle f|g\rangle$,

$(\Delta A)^{2}(\Delta B)^{2}\geq \left({\frac {\langle f|g\rangle -\langle g|f\rangle }{2i}}\right)^{2}$.

But

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle f|g\rangle = \langle(\hat{A}-\langle A\rangle)\Psi|(\hat{B}-\langle B\rangle)\Psi\rangle = \langle\Psi|(\hat{A}-\langle A\rangle)(\hat{B}-\langle B\rangle)\Psi\rangle }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \langle\Psi| \hat{A}\hat{B} - \hat{A}\langle B\rangle - \hat{B}\langle A\rangle + \langle A\rangle\langle B\rangle)\Psi\rangle }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \langle\Psi| \hat{A}\hat{B}\Psi\rangle - \langle B\rangle\langle\Psi|\hat{A}\Psi\rangle - \langle A\rangle\langle\Psi|\hat{B}\Psi\rangle + \langle A\rangle\langle B\rangle\langle\Psi|\Psi\rangle }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle\hat{A}\hat{B}\rangle - \langle B\rangle\langle A\rangle - \langle A\rangle\langle B\rangle + \langle A\rangle\langle B\rangle = \langle \hat{A}\hat{B}\rangle - \langle A\rangle\langle B\rangle } .

Similarly,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle g|f\rangle = \langle\hat{B}\hat{A}\rangle - \langle A\rangle\langle B\rangle }

so

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle f|g\rangle - \langle g|f\rangle = \langle\hat{A}\hat{B}\rangle - \langle\hat{B}\hat{A}\rangle = \langle[\hat{A},\hat{B}]\rangle }

where

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [\hat{A},\hat{B}] \equiv \hat{A}\hat{B} - \hat{B}\hat{A} }

is the commutator of the two operators.

Conclusion:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\Delta A)^2(\Delta B)^2 \geq \left (\frac{1}{2i}\langle[\hat{A},\hat{B}]\rangle\right )^2} .

This is the generalized uncertainty principle.

So, here is a true physical constraint on a wave packet. If we compress it in one variable, it expands in the other! If it is compressed in position (i.e., localized) then it must must be spread out in wavelength. If it is compressed in momentum, it is spread out in space.

So now let's see what is the physical meaning of Heisenberg’s inequalities?

1. Suppose we prepare N systems in the same state. For half of them, we measure their positions x; for the other half, we measure their momenta Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_{x}} . Whatever way we prepare the state of these systems, the dispersions obey these inequalities.

2. These are intrinsic properties of the quantum description of the state of a particle.

3. Heisenberg uncertainty relations have nothing to do with the accuracy of measurements. Each measurement is done with as great an accuracy as one wishes. They have nothing to do with the perturbation that a measurement causes to a system, inasmuch as each particle is measured only once.

4. In other words, the position and momentum of a particle are defined numerically only within limits that obey these inequalities. There exists some “fuzziness” in the numerical definition of these two physical quantities. If we prepare particles all at the same point, they will have very different velocities. If we prepare particles with a well-defined velocity, then they will be spread out in a large region of space.

5. Newton’s starting point must be abandoned. One cannot speak simultaneously of x and p. The starting point of classical mechanics is destroyed. Some comments are in order.

A plane wave corresponds to the limit Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta p\rightarrow 0} . Then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta x} is infinite. In an interference experiment, the beam, which is well defined in momentum, is spread out in position. The atoms pass through both slits at the same time.We cannot “aim” at one of the slits and observe interferences.

The classical limit (i.e. how does this relate to classical physics) can be seen in a variety of ways that are more or less equivalent. One possibility is that the orders of magnitude of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} are so large that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\hbar}{2}} is not a realistic constraint. This is the case for macroscopic systems. Another possibility is that the accuracy of the measuring devices is such that one cannot detect the quantum dispersions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta x} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta p} .

See also Generalized Heisenberg Uncertainty Relation

As a trivial example, suppose the first observable is position Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\hat{A}=\hat{x})} , and the second is momentum Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{B}=\hat{p}=\frac{\hbar}{i}\frac{d}{dx}} .

The commutation relation between these two observables is just

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [\hat{x},\hat{p}]=i\hbar } .

So,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\Delta x)^2 (\Delta p)^2 \geq \left (\frac{1}{2i}\cdot i\hbar\right )^2 = \left (\frac{\hbar}{2}\right )^2 } .

A worked problem showing the uncertainty in the position of different objects over the lifetime of the universe: Problem 1

A problem about how to find kinetic energy of a particle, a nucleon specifically, using the uncertanity principle : Problem 2

Another problem verifying Uncertainty relation: Problem 3